On Shallit’s Minimization Problem

Abstract

In Shallit’s problem (SIAM Review, 1994), it was proposed to justify a two-term asymptotics of the minimum of a rational function of $n$ variables defined as the sum of a special form whose number of terms is of order $n^2$ as $n\to\infty$. Of particular interest is the second term of this asymptotics (“Shallit’s constant”). The solution published in SIAM Review presented an iteration algorithm for calculating this constant, which contained some auxiliary sequences with certain properties of monotonicity. However, a rigorous justification of the properties, necessary to assert the convergence of the iteration process, was replaced by a reference to numerical data. In the present paper, the gaps in the proof are filled on the basis of an analysis of the trajectories of a two-dimensional dynamical system with discrete time corresponding to the minimum points of $n$-sums. In addition, a sharp exponential estimate of the remainder in Shallit’s asymptotic formula is obtained.